Question: Simplify the following expression: $t = \dfrac{81x^2 - 27x}{-81x^2 - 9x}$ You can assume $x \neq 0$.
Answer: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $81x^2 - 27x = (3\cdot3\cdot3\cdot3 \cdot x \cdot x) - (3\cdot3\cdot3 \cdot x)$ The denominator can be factored: $-81x^2 - 9x = - (3\cdot3\cdot3\cdot3 \cdot x \cdot x) - (3\cdot3 \cdot x)$ The greatest common factor of all the terms is $9x$ Factoring out $9x$ gives us: $t = \dfrac{(9x)(9x - 3)}{(9x)(-9x - 1)}$ Dividing both the numerator and denominator by $9x$ gives: $t = \dfrac{9x - 3}{-9x - 1}$